Recent questions in FTOC

Integral CalculusAnswered question

grenivkah3z 2022-07-08

What is the difference between first and second fundamental theorem of calculus?

Integral CalculusAnswered question

Sam Hardin 2022-07-05

Given a continuous function $f:[0,1]\to \mathbb{R}$, prove that

$\mathrm{\forall}t>0,\frac{1}{t}\cdot \mathrm{ln}\left({\int}_{0}^{1}{e}^{-tf(x)}dx\right)\le -minf(x).$

$\mathrm{\forall}t>0,\frac{1}{t}\cdot \mathrm{ln}\left({\int}_{0}^{1}{e}^{-tf(x)}dx\right)\le -minf(x).$

Integral CalculusAnswered question

kolutastmr 2022-07-02

Finding $f(x)$ in ${\mathrm{cos}}^{2}(x)f(x)={x}^{2}-2{\int}_{1}^{x}\mathrm{sin}(t)\mathrm{cos}(t)f(t)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t$

Integral CalculusAnswered question

Crystal Wheeler 2022-07-01

How do you use the Fundamental Theorem of Calculus to find the derivative of $\int \frac{1}{1+{t}^{2}}dt$ from $x$ to $5$?

Integral CalculusAnswered question

pipantasi4 2022-07-01

Finding the maximum of $G(x)={\int}_{x}^{x+a}f(t)\phantom{\rule{thinmathspace}{0ex}}dt$ using FTOC.

Integral CalculusAnswered question

lilmoore11p8 2022-07-01

Find the derivative of the following function using the Fundamental Theorem of Calculus:

$F(x)={\int}_{{x}^{3}}^{{x}^{6}}(2t-1{)}^{3}dt$

$F(x)={\int}_{{x}^{3}}^{{x}^{6}}(2t-1{)}^{3}dt$

Integral CalculusAnswered question

glitinosim3 2022-07-01

Let $f$ be continuous on $I=[a,b]$ and let $H:I\to \mathbb{R}$ be defined by $H(x)={\int}_{x}^{b}f(t)dt,x\in I$. Find ${H}^{\prime}(x)$.

Integral CalculusAnswered question

Leland Morrow 2022-07-01

Can you apply the fundamental theorem of calculus with the variable inside the integrand?

For example: $\frac{d}{dx}\left({\int}_{a}^{x}xf(t)dt\right)$

For example: $\frac{d}{dx}\left({\int}_{a}^{x}xf(t)dt\right)$

Integral CalculusAnswered question

Sarai Davenport 2022-06-28

What is the difference between the two parts of FTOC?

Integral CalculusAnswered question

oleifere45 2022-06-25

How does the first fundamental theorem of calculus guarantee the existence of antiderivatives of functions?

Integral CalculusAnswered question

Emanuel Keith 2022-06-25

$f(x)={\int}_{0}^{\mathrm{sin}x}1+\mathrm{sin}(\mathrm{sin}(t))dt$

Find $({f}^{-1}{)}^{\prime}(0)$

Find $({f}^{-1}{)}^{\prime}(0)$

Integral CalculusAnswered question

excluderho 2022-06-24

Derivative of a function and an integral

$\frac{d}{dx}({x}^{6}({\int}_{0}^{sinx}\sqrt{t}dt))$

$\frac{d}{dx}({x}^{6}({\int}_{0}^{sinx}\sqrt{t}dt))$

Integral CalculusAnswered question

Yahir Tucker 2022-06-24

We know if $g$ is continuous on $(a,b)$ and $F(x)={\int}_{a}^{x}g(t)dt$, then

${F}^{\prime}(x)=g(x)$

But, how about if we have

$F(x)={\int}_{a}^{h(x)}g(t)dt$

What should ${F}^{\prime}(x)$ be?? can we still apply fundamental theorem of calculus?

${F}^{\prime}(x)=g(x)$

But, how about if we have

$F(x)={\int}_{a}^{h(x)}g(t)dt$

What should ${F}^{\prime}(x)$ be?? can we still apply fundamental theorem of calculus?

Integral CalculusAnswered question

Petrovcic2x 2022-06-22

Can use FToC to evaluate $\underset{x\to \mathrm{\infty}}{lim}\frac{{\int}_{0}^{x}\phantom{\rule{mediummathspace}{0ex}}f\left(t\right)dt}{{x}^{2}}$?

Integral CalculusAnswered question

Poftethef9t 2022-06-20

If $f(x)$ is even, then what can we say about:

${\int}_{-2}^{2}f(x)dx$

If $f(x)$ is odd, then what can we say about

${\int}_{-2}^{2}f(x)dx$

Are they both zero? For the first one if its even wouldn't this be the same as

${\int}_{a}^{a}f(x)dx=0$

Now if its odd $f(-x)=-f(x)$. Would FTOC make this zero as well?

${\int}_{-2}^{2}f(x)dx$

If $f(x)$ is odd, then what can we say about

${\int}_{-2}^{2}f(x)dx$

Are they both zero? For the first one if its even wouldn't this be the same as

${\int}_{a}^{a}f(x)dx=0$

Now if its odd $f(-x)=-f(x)$. Would FTOC make this zero as well?

Integral CalculusAnswered question

watch5826c 2022-06-19

Show that $g$ is differentiable and find ${g}^{\prime}(x)$, FTOC

Integral CalculusAnswered question

Carolyn Beck 2022-06-17

Find the derivative of integral

$H(x)={\int}_{3}^{{\int}_{1}^{x}{\mathrm{sin}}^{3}tdt}\frac{dt}{1+{t}^{2}+{\mathrm{sin}}^{6}t}$

$H(x)={\int}_{3}^{{\int}_{1}^{x}{\mathrm{sin}}^{3}tdt}\frac{dt}{1+{t}^{2}+{\mathrm{sin}}^{6}t}$

Integral CalculusAnswered question

juanberrio8a 2022-06-15

Prove

${\int}_{-a}^{a}f(x)dx=0$

assuming $f(x)$ is odd.

${\int}_{-a}^{a}f(x)dx=0$

assuming $f(x)$ is odd.

Integral CalculusAnswered question

Poftethef9t 2022-06-11

Why does FTOC apply here, to find the derivative of ${\int}_{\mathrm{sin}(x)}^{\pi}\frac{t}{\mathrm{cos}(t)}dt$

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